Problem 10 - Intermediate Algebra

Question

For Problems \(1-18\), write each of the following in scientific notation. \(9126.21\)

Step-by-Step Solution

Answer

The number 9126.21 in scientific notation is \(9.12621 \times 10^3\).

Step 1: Understanding Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small conveniently. It is written in the form of \(a \times 10^n\), where \(1 \leq a < 10\), and \(n\) is an integer.

Step 2: Identify the Coefficient

We need to adjust 9126.21 to create a number between 1 and 10. We do this by moving the decimal point left until we have 9.12621, because we need a single non-zero digit to the left of the decimal.

Step 3: Determine the Exponent

Count how many places the decimal point moved to the left to transform 9126.21 into 9.12621. Since the decimal moved 3 places to the left, the exponent \(n\) will be 3.

Step 4: Write in Scientific Notation

Combine the coefficient and the exponent to write the number in scientific notation. Since the coefficient is 9.12621 and the exponent is 3, the scientific notation is \(9.12621 \times 10^3\).

Key Concepts

Mathematics Education

Mathematics education involves teaching and learning concepts that help us understand numbers and their applications better. One integral part of this is scientific notation. This mathematical system is crucial because it allows students to grasp the basics of handling very large or very small numbers efficiently. By expressing numbers in the form of \(a \times 10^n\), students learn to simplify complex numerical expressions.
This format not only simplifies calculations but also aids in understanding the significance of numbers in various contexts. By mastering scientific notation, learners cultivate problem-solving skills necessary for higher-level mathematics and science courses. Reinforcing these concepts through examples and exercises helps build a strong foundation in numerical literacy.

Expressing Large Numbers

Expressing large numbers can sometimes feel overwhelming, but scientific notation offers a neat solution. Take the example of converting the number 9126.21 into scientific notation. The goal is to make the number more manageable by adjusting it into a format that is easier to read and work with.
To start, we need to identify a single non-zero digit to the left of the decimal point. In the number 9126.21, we move the decimal point to get 9.12621. This ensures that we have a coefficient \(a\) where \(1 \leq a < 10\).
Scientific notation thus transforms bulky numbers into a streamlined form, making them user-friendly especially in subjects like physics and engineering. With this method, numerical data becomes simpler to handle across various fields.

Decimal Point Movement

Understanding decimal point movement is key when converting numbers into scientific notation. Decimal point movement decides the value of the exponent in \(10^n\). For the number 9126.21, we moved the decimal point three places to the left to form 9.12621.
The places the decimal point moves correspond directly to the exponent \(n\).

• When the decimal is moved to the left, the exponent is positive.
• Conversely, when moved to the right, the exponent is negative, used when dealing with numbers less than one.

By getting the hang of how decimal point movement works, you can thus easily convert any number into a scientific notation format. It's an essential skill that simplifies dealing with extensive or minuscule numbers efficiently.